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Oil  a  Certain  Class  of  Functions 
with  Line-Singularities 


DISSERTATION  PRESENTED  TO  THE  BOARD  OP  LNIVERSITY 

STUDIES  OF  THE  JOHNS  HOPKINS  UNIVERSITY  POR 

THE  DEGREE  OP  DOCTOR  OP  PHILOSOPHY 


BY 


JOHN  blbSLAiXD 


BALTIMORE,  1898 


BASTON,   PA.  : 

THE  CHEMICAL  PUBLISBIKG  COMPANY. 

1899. 


On  a  Certain  Class  of  Functions 
with  Line-Singularities 


DISSEKTATIOX  I'KESEXTED  TO  THE  BOAKD  OF  LMVEKSITY 

STUDIES  OF  THE  JOHNS  HOPKINS  UNIVEKSITY  FOK 

THE  DEGKEE  OF  DOCTOR  OF  FHILOSOFHY 


BY 


J0| 

UNIVERSITY 

BALTIMORE,  1898 


EABTON,   PA.  : 

THE  CHEMICAL  PUBLISHING  COMPANY. 

1899. 


^ 


INTRODUCTION. 

The  object  of  this  paper  is  to  investigate  certain  properties 
of  functions  with  line-singularities.  In  the  first  part  a  class  of 
functions  of  a  real  variable  has  been  considered  which  are  closely 
connected  with  these  transcendentals,  and  in  the  case  of  which 
MacLaurin's  development  does  not  hold,  in  spite  of  the  fact  that 
all  the  derivatives  of  the  function  are  finite  as  well  as  the  func- 
tion itself.  Such  functions  were  considered  by  Pringsheim  in 
an  article  "Zur  Theorie  der  Taylorschen  Reihe,"  Math.  Ann.^ 
Vol.  42.  Attention  has  been  called  to  certain  functions  of  this 
kind  which  all  give  the  same  development  in  a  MacLaurin's 
series,  the  identity  of  the  different  series  being  due  to  the 
periodicity  of  the  coefficients.  The  correspondence  between 
these  functions  and  the  functions  represented  by  their  expansion 
in  series  has  been  shown,  and  also  how  Poincar^'s  method  of 
solving  an  infinite  system  of  equations  with  an  infinite  number 
of  imknowns  may  be  applied  in  expanding  such  functions. 
The  above-mentioned  correspondence  was  pointed  out  by  Borel 
in  his  thesis,  "  Sur  Quelques  Points  de  la  Theorie  des  Func- 
tions," Paris,  1894. 

In  the  second  part  functions  of  the  type 


/w=2'.- 


have  been  considered,  a^  being  an  ensemble  forming  at  most 
essentially  singular  lines.  Different  methods  of  developing  in 
series  have  been  discussed,  development  in  areas  bounded  by 
ellipses  and  other  curves,  these  being  essentially  singular  lines.  It 
is  also  shown  how  BorePs  theorem  may  be  applied  to  such  func- 
tions, and  that  they  may  be  continued  across  the  singular  lines 
without  loss  of  continuity,  provided  the  variable  is  restricted  to 
move  in  certain  directions. 

The  third  part  has  been  devoted  to  the  study  of  functions 
possessing  the  property  of  remaining  finite  and  continuous  as 


357063 


well  as  all  its  derivatives  on  the  singular  line.  It  was  Prof. 
Mittag-Leffler  who  first  called  the  attention  of  mathematicians 
to  this  class  of  functions  in  a  note  in  Ac^a  Math.^  Vol.  XV, 
where  he  gives  an  example  of  such  a  function  constructed  by 
his  pupil  F'redholm.  Pringsheim,  in  the  above-mentioned  paper, 
discusses  functions  possessing  a  similar  property,  and  Borel,  in 
his  thesis,  gives  another  method  of  generating  these  functions. 
A  third  method  was  discovered  during  the  course  of  my  work, 
in  fact,  I  have  shown  that  functions  of  the  form 

-^   Z    by  , 

where  ^A^)  =  — ^ — ; — 7-^\ — :•  +  ^     — ^ r-r^ — ;■  1    +  •  •  • 

can  by  proper  choice  of  ensembles  a^  and  d^,  be  made  to  possess 
the  above-mentioned  property.  Another  simpler  type  included 
in  the  more  general  one  given  above,  z/tz. : 

;/=  I 

has  been  shown  to  be  closely  related  to  the  form  given  by 
Pringsheim, 

I  am  indebted  to  Prof.  A.  Chessin  for  valuable  suggestions 
and  criticisms  offered  diiring  the  final  revision  of  this  paper. 

I. 

We  shall  call  a  function  holomorphic  in  a  region,  if  at  all 
points  within  it  the  function  is  finite  and  continuous  and  pos- 
sesses a  finite  and  continuous  derivative.  At  any  point  within 
such  a  region  the  function  may  be  represented  by  a  convergent 
powerseries  V{x  —  x^^  having  a  radius  of  convergence    greater 


than  zero.  Suppose  now  that  x^  moves  continuously  from  a 
position  x^  to  X.  If  the  path  does  not  pass  through  a  critical 
point,  there  will  to  any  position  of  x^^  say  X',  correspond  a  de- 
velopment of  the  function  in  a  powerseries  F{x  —  X')  with  a 
radius  of  convergence  greater  than  zero.  From  this  series  we 
may,  by  the  so-called  analytical  continuation,  derive  a  new  series 
representing  the  function  at  a  point  taken  within  the  circle  of 
convergence  of  P(:r —  X'),  and  so  on.  The  particular  expression 
for  the  function  at  any  point  in  the  path  is  called  by  Weierstrass 
an  element  of  the  function,  and  the  totality  of  these  elements 
define  a  monogenic  and  analytic  function  in  that  part  of  the 
plane  which  is  covered  by  the  circles  of  convergence. 

The  great  majority  of  analytic  functions  exist  in  general 
throughout  the  whole  plane,  except  at  certain  points,  the  singu- 
lar points.  These  may  be  branch-points,  polar,  logarithmic,  or 
essential  singularities. 

The  concept  of  an  analytic  function  necessarily  involves 
the  property  of  possessing  finite  and  determinate  derivatives 
without  which  no  development  is  possible.  That  finite  and 
continuous  functions  do  exist  which  have  no  derivatives,  and 
thus  cannot  be  represented  as  an  analytic  function,  Weierstrass 
has  proved  beyond  any  doubt'  In  fact,  it  is  well-known  that  the 
function  represented  by  the  series 


g{2>t=^b-z-\ 


where  3  is  a  positive  quantity  less  than  unity  and  a  a  positive 
odd  integer,  converges  unifonnly  and  unconditionally  for  all 
points  within  and  on  the  circle  \z\  =  i  and  diverges  for  all 
points  outside;  this  function,  moreover,  does  not  possess  any 
derivatives  on  the  circle  of  convergence,  if  ^(5  >  i  -(-  f  tt, 
although  it  is  holomorphic  for  all  points  within.  If  we  put 
z  =  e*'  and  take  the  real  part  of  the  series,  we  get  the  function 

00 


'  See  Weierstrass.  Abh.  aus  der  Funktionenlehre,  p.  91. 


which  is  a  convergent  series  and  represents  a  finite  and  continu- 
ous function  of  a  real  variable.  Weierstrass  has  shown  that  this 
function  for  no  value  of  6  possesses  a  determinate  differential 
coefficient,  provided  «<^  >  i  +  |  tt.     By  means  of  the  function 


<i>{x)  ==  E(^)  +  yx—'E{x), 

where  E(:r)  stands  for  the  greatest  integer  in  x^  Schwarz'  has 
constructed  a  function  which  is  finite  and  continuous  and  pos- 
sessing nowhere  a  derivative.  This  function  is  given  by  the 
series 


Ax)=^n 


<I>{2"X) 
2".  2" 


We  may  suspect  that  an  indefinite  number  of  such  functions 
exist,  which  in  fact  is  the  case,  as  has  been  shown  by  Lerch, 
who  has  constructed  a  powerseries  including  Weierstrass'  series 
as  a  special  case  (Crelle,  Vol.  CIII). 

The  property  of  possessing  finite  and  determinate  deriva- 
tives is  however  not  sufficient  to  make  sure  that  the  function  is 
analytic.  In  fact,  given  the  analytic  function  /[x)  on  one  side, 
and  on  the  other  the  Taylor  series 

the  question  of  convergence  of  this  series  naturally  presents 
itself,  and  even  if  the  Taylor  series  is  convergent,  we  are  a 
priori  not  at  all  certain  that  the  development  so  formed  repre- 
sents the  function  f{pc)  itself.  Cauchy  as  early  as  1823  ^d" 
vanced  this  idea  in  opposition  to  Lagrange,  who  claimed  that 
the  finiteness  and  continuity  of  f"  {x)  was  enough  to  establish 
the  convergence  and  validity  of  the  expansion  given  above. 
The  example  used  by  Cauchy  to  illustrate  this  was,  however, 
not  a  happy  one,  and  some  mathematicians^  consider  it  insuffi- 
cient to  prove  the  statement  he  made  concerning  the  validity  of 

'  Schwarz,  Gesammelte  Abhandlungen,  II,  pp.  279-274. 
"^  See  Pringsheim,  "Zur  Theorie  der  Taylorschen  Reihe, ' '  Math.  Annalen, 
B.  42,  pp.  155  and  161. 


an  expansion.  The  example  given  by  Pringsheim  in  the  above- 
mentioned  paper  illustrates  clearly  what  we  have  said.  In  fact, 
the  function 

when  expanded,  gives  rise  to  the  series 

which  is  convergent  throughout  the  finite  portion  of  the  plane. 
That  4>{^)  does  not  represent  /{x)  is  evident  from  the  fact  that 
/{x)  in  the  neighborhood  of  ;r  =  o  has  an  infinite  number  of 
poles,  X  ^  o  being  a  point-limit  of  poles,  while  <^{x)  is  holo- 
morphic  throughout  the  whole  plane. 

We  shall  now  proceed  to  give  a  few  examples  which  enable 
us  to  put  directly  into  evidence  the  non-identity  of  the  expan- 
sion with  the  function  expanded  without  resorting  to  function — 
theoretical  considerations.  Pringsheim's  attempt  in  this  direc- 
tion does  not  seem  to  be  a  complete  success,  on  account  of  the 
choice  of  the  function  necessitating  a  laborious  method  of  ap- 
proximation.    Consider  the  series 


/(-)-=2 


(2v-fi)/      I  +  a»''  +  'A:' 


where  a'ls  a.  positive  integer  greater  than  one.     The  series  is 
uniformly  convergent  for  all  values  of  x  with  the  exception  of 

the  points  —  — ,  -\ j^  . . . ,  situated  on  the  negative  half  of 

the  .r-axis,  zero  being  a  point-limit  as  before.     We  find 

/  KX)       ^      U«.^     (2K+  I)/    (i+a»'+'Ar)''+'  ' 

o 


(2V+I)./ 


=  Sin  X, 


■^     ^  ^  ^     (2V+  I)/  ^  ^ 

and  we  arrive  at  the  following  expansion : 

<l>{x)  =  sin  IT  —  sin  aw.x  -{-  sin  a^ir.x^  —  .... 

Suppose  now  a  =  a.  positive  integer  greater  than  unity,  each  term 

in  this  series  vanishes.     We  have  thus  a  function  with  the  same 

I 
property  as  Cauchy's  well-known  /{x)  =  e   "^,    but  free  from 

all  the  objections  that  may  be  raised  against  this  example.     It  is 

also  much  simpler  than  the  one  given  by  Pringsheim'.     That  (t>{x) 

does  not  represent  y"(;c)  is  obvious  from  the  fact  thaty(;ir)  is  zero 

for  ;r  =  o,  and  differs  from  zero  for  all  other  values  of  x  in  the 

neighborhood  of  the  origin.^     If  instead  of  ir  we  write  — ,  /{x) 
becomes 

and  the  corresponding  expansion  is 

,  -    ,          .      TT          .air       .     .    a^TT      „ 
<t>{x)  =  sm sm  —  .x-\-s\n .x'  —  . . ., 

2  2  2 

a  being  subject  to  the  same  condition  as  before.     If  a  is  an 
even  integer,  we  get 

4>^{x)  =  I, 

and  when  a  is  an  odd  integer  of  the  form  4«  +  i,  we  have 

<i>.Xx)  =1  —  x -\- x'  —  x^-\-  ..•, 

and  for  a  =  an  odd  integer  of  the  form  4/2  —  i 


'  Pringsheim's  example  is 

/(.)  =2'  <-)'  {  ™  T+W  -  (t)  "'-"  }  • 

^  The  non-identity  of/(jf)  and  <P(x)  is  also  evident,  if  we  apply  the  well- 
known  theorem  that  a  function  cannot  be  identically  zero  in  any  finite  part  of 
the  plane,  or  at  all  points  on  a  line  of  finite  length,  without  being  identically 
zero  throughout  the  whole  plane. 


Neither  of  these  expansions  represents  the  respective  functions. 
In  fact,  <^j(jr)=  i  =  const.,  while  f{x^  2n)  is  not  a  constant  for 

all  values  of  x.     In  the  same  way,  <t>Jix)  =  — -p —  and  4>^{x)  = 

I  -\-  X 

The  first  case  illustrates  the  singular  fact  that  func- 


I 


I  —X 

tions  do  exist,  all  whose  derivatives  vanish  at  a  given  point, 

without  the  functions  being  constant.  In  the  preceding  exam- 
ple the  constant  happened  to  be  zero.  We  might  easily  multi- 
ply examples  of  this  kind ;  thus  the  function 


/<'>-2'^^(i)"TT^ 


-\-a^''x 

o 

gives  rise  to  the  expansion 

<t>{x)  =  cos cos — .  x-\-  cos  — .  X  —  ... 

2  2  2 

and  making  different  hypotheses  concerning  the  constant  a  we 
arrive  at  results  similar  to  the  above.  Again,  we  may  distribute 
the  poles  on  the  positive  half  of  the  ;r-axis  and  subtract  the  re- 
sulting function  from  the  one  already  considered ;  the  new  func- 
tion will  present  the  same  features  as  the  above..  We  may  also 
distribute  the  poles  on  the  imaginary  j-axis  and  thus  derive 
functions  similar  to  the  above.  Finally  we  notice  this  impor- 
tant fact,  to  the  holomorphic  function  f{x)  corresponds  some 
function  <f>{x),  but  the  correspondence  is  not  a  one  to  one,  in  fact, 
to  a  function  (f>{x)  corresponds  an  infinite  number  of  functions 
/"(;r),  while  to  f{x)  corresponds  only  one  holomorphic  function. 
This  is  entirely  due  to  the  periodicity  of  the  coefficients  in  the 
expansion.  These  examples  also  illustrate  the  fact  that  a  sum 
of  Taylor's  series  whose  radii  of  convergence  tend  to  zero  may 
have  a  finite  and  even  infinite  radius  of  convergence. 

*  The  non-identity  of  these  expansions  may  be  proved  more  rigorously 
thus  :    take  the  first  case.     If  we  can  prove  the  non-identity  of  'Pi(x)  and 

^{x,  2ff )  when  x  takes  any  f>ositive  value  >  —  it  is  evident  that  the  non-iden- 
tity is  proved  for  all  values  of  ;r.  Wehave/(jr,  2«;< -^ <  i,  {^> J» 

while  ^i(^)  =  I.     The  value  ;r  =o  is  of  course  excluded. 


lO 

We  mentioned  above  that  there  exists  a  correspondence 
between  certain  functions  of  the  type  fix)  =  ^  ■  "  ^  ,  hav- 
ing different  sets  of  poles,  and  their  corresponding  expansion 
due  to  the  periodicity  of  the  coefficients  of  the  expansion.  We 
shall  now  show  that  a  correspondence  of  a  somewhat  similar 
nature  exists  in  the  case  of  functions  of  the  same  type,  all  hav- 
ing the  same  poles,  but  different  sets  of  coefficients  c^.^ 
Let 

where  a„  is  an  ensemble,  such  that  lim.  /2^  =  oo  .     In  order  that 

this  function  shall  be  of  the  kind  we  have  been  discussing,  we 
must  have 

(i)     ^p c^  =  A„,  ^,>- Cytty  =* A.,    ...,  ^M <:,«."  =  A„, 

o  o  o 

where  the  A's  are  fixed  quantities,  such  that  the  series 

<^{x)  =A„  — A,;ir-|- A,;*;'' 

shall  be  absolutely  convergent.  If  then  solutions  c^  of  the  sys- 
tem (i)  can  be  found,  then  will  f{pc)  have  the  required  property. 
But,  as  Poincare  has  shown,^  we  can  always  solve  such  a  system, 
and  since  these  solutions  are  not  unique,  it  follows  that  a  single 
development  ^{x)  may  be  derived  from  an  infinite  number  of 
different  functions  fix).  As  an  example  let  us  take  the  func- 
tion 


•^(•^)=2''TT^' 


o 

where  «  is  a  fixed  real  quantity  >  i.     Putting  all  the  A's  equal 
to  unity  our  system  (i)  becomes 


o,  ^c,a-  —  I"  -=  o,   . . . ,  ^/.g"^ 


o. 


'  The  correspondence  of  this  latter  kind  has  been  pointed  out  by  Borel 
in  the  above-mentioned  thesis. 

*  Bulletin  de  la  Society  Math^matique  de  France,  Tome  VIII. 


II 


In  order  to  solve  this  system  we  proceed  by  Poincar^'s  method  ; 
we  fonn  a  function  whose  zeros  are  the  points  —  i,  +  i,  a,  rt", 
.  . . ,  viz.  : 

FU)  ==  (I -;.:)  (I  +  ;«;)  IJ  (i  -  ^), 

I-  =  I 

a  function  of  genus  zero,  since  ^S —  is  convergent.      We  now 

draw  an  infinite  system  of  concentric  circles  C,  C„,  C,,  C^,  . . . , 
C^,  . . . ,  having  the  origin  as  center  and  such  that  no  circle 
passes  through  a  zero-point  of  the  function  V{x\  If  now  the 
integral 

\    x^dx 


F(;i:) 

JCv 


taken  around  a  circle  C^  constantly  approaches  zero,  whatever 
be  the  value  of  ;/,  as  the  circle  of  integration  becomes  indefi- 
nitely large,  it  is  evident  that  we  must  have 

00 

(2)  ^.'B^'"'—  BCO-^o,  «  =  o,  I,    ...,00. 

o 

where  the  B's  are  the  residues  of  the  function  — with  respect 

F(;»r) 

to  the  poles  —  i,  i,  «,  «',  .... 
These  residues  are 


F'(-i)       '       F'(i)'      '       F'(a)'**"'  F'(a  )  • 

and  we  readily  find  the  following  values  for  the  B's, 

B  ^ ,    B»  = >  *  *  • » 

n  (.+--■,)     .n(>-i.) 


B  _  ^  

'<'-«^)irw(.-5) 


12 

where  the  product  sign  ITj^)  means  that  the  factor  in  which 
V  ^  r  is  absent.  These  residues  give  at  once  the  required  solu- 
tions of  our  system.     In  fact,  it  is  only  necessary  to  put 

P  _^    P  — Jii  c  — 

^0        B  '      '~   B  '    *"'      "        B  '    *** 

in  order  to  satisfy  the  system  (i).  It  remains  now  to  prove  that 
the  integral 

I     x^dx 


approaches  zero  for  v  =  oo .     To  do  this  we  put 

F{x)  =  {i  ~x){i+x)  f]  (i  -^)  =--  (I  -^)(i  +  x)^(x). 

V  =  I 

We  have 

,      ._|  r x"clx I       r l^^ll^^l 

'  ""'"I  J(i-^)(i+^)<l'(^)l^  Jl(i-^)  (1+^11^(^)1 

_^  r\x"\  \dx\: 
^  k   J      \^{x)\ 

where  k  is  the  smallest  value  that    1 1  —  ;i:  I  1 1  +  :r  |  can  have  on 
C;,.     We  have  thus 

.r«  1 1  flfx  I 


where  M^  is  the  smallest  value  of    |<I>(:^')|  on  C^.     But  putting 

,  \\X''\  \dx\   _27rp,"+^ 

where  py  is  the  radius  of  C^,  we  have 

I  I  —  p..   I 

Let  us  now  suppose  that  the  circles  C^  are  drawn  in  such  a  way 
as  to  satisfy  the  equation 


13 

which  evidently  can  be  done  without  ever  having  any  circle 
pass  through  a  zero  of  F(a:).     We  have  also 

<^{ax)  =  {i — x)^{x), 
so  that  we  get 

_  \\x"\\dx\_        I  27r/j,«  +  'a''  +  '_  a^  +  'J.^ 

Jk  +  I.i.  


M„+,  \l   —   py\  M„'  \l—pA     ' 


which  gives  at  once 


=  o  for  V  =  00 


This  equation  shows  that  J^  „  must  approach  zero,  when  v 
becomes  infinite  and  therefore  also  I^.„, — q.  e.  d. 

We  have  thus  arrived  at  the  following  simple,  but  false, 
expansion  of  y^(^), 

<f>{x)  =  I  — x~\-x*  —  . . ., 

which  is  convergent  within  the  unit  circle.    The  solutions  of  the 
system  (i)  are  not  unique,  in  fact,  the  series 

c^a\  c^a,  c,a*,   ....  c^/i",  —  i 

are  equally  well  solutions  of  (i),  and  more  generally,  c^a^^,  c^a^y 

,  r^"^,  . . .,  ( —  i)*",  are  solutions  of  the  system  ;    but  to  all 

the  different  functions  /{x)  that  may  thus  be  constructed  there 
corresponds  only  one  expansion  <t>{x). 


II. 

We  shall  now  take  up  the  study  of  a  more  general  type  of 
functions  analogous  to  the  one  discussed  above.     Let 


<■>  /<->=2'r^ 


in  which  2  U J  is  an  absolutely  convergent  series,  and  we  shall 
suppose  a^y  a^,  . . . ,  «^,  ...  to  be  some  enumerable  ensemble  hav- 
ing at  least  one  point-limit  not  belonging  to  the  ensemble  itself, 
or,  to  use  Cantor's  definition,  our  ensemble  is  not  perfect.    We 


H 

shall  further  suppose  the  points  a^,  to  be  condensed  at  most  on 
lines  and  not  in  an  area.  Let  a^  be  some  point  of  the  ensemble 
and  let  a  circle  with  radius  R  and  center  r^  be  drawn  through 
this  point  and  such  that  no  other  point  of  the  ensemble  lies 
within  the  circle."  We  shall  prove  the  following  theorem  due 
to  Poincare :  ^ 

T/ie  series  f{x)  represejits  inside  of  the  circle  of  radius  R^ 
center  x^  =  o,  a  monogenic  analytic  function^  and  therefore 
developable  in  a  powerseries  convergent  for  \x\  <C.R  and  diver, 
gent  for  \x\  y.  R.  We  shall  follow  Goursat's  exposition  of  the 
proof.  3 

We  have  by  hypothesis 

|a„l  =R,    ...,    la.l  >R,  v  >  o. 

If  we  give  to  x  an  absolute  value  v  <  R,  we  may  develop 
each  term  of  fix)  in  a  convergent  powerseries 

f^x-)  =  ~^^^—  =  ^-\-'^^x^^,x'^...^-\^x-^..., 
a^  —  x        a^    ^    a"'       'a/  '     a/'  +  i  ' 

K=  o,  I,  2,  ....      But  f{x)  considered  as  the  sum  of  all  these 

series  is  finite  and  less  than  :^  >^  kJ,    and  therefore  the 

r — K^^iLJ 

order  in  which  we  sum  the  series  is  indifferent.      Addinor  then 

by  vertical  columns  we  get 

(2)  /(;^)_A„+A.;t4---.  +  A„;ir«+..., 

where 

OC  00  00 

A„^5!  — .    A,=^.4,   •..,  A.=  ^.-^,   ... 

o  o  o 

f(x)  is  therefore  monogenic  and  analytic  within  the  circle  C  of 

'  An  ensemble  is  said  to  be  condensed  in  an  interval  a,  •  •  • ,  5,  if  an  inter- 
val j3,  .  • . ,  7,  as  small  as  we  please,  contained  in  a,  •  •  • ,  5,  contains  points  of  the 
ensemble.     See  article  by  G.  Cantor,  Acta  Math.,  Vol.  2,  p.  351. 

*  Acta  Soc.  Fenn.,  Tome  XII  (1883),  pp.  341-350. 

^  Sur  les  Fonctions  a  Espaces  Lacunaires,  Bulletin  des  Sciences  Math., 
1887,  2*""^  Serie,  Tome  XI,  p.  109. 


15 


radius  R,  having  its  center  at  the  point  x^  =  o.  Further,  the 
series  (2)  is  divergent  if  |;r|^  R.  In  fact,  since  ^^  \Cy\  is  con- 
vergent it  is  possible  to  find  a  number  n  such  that  we  shall  have 

2kvi  <o\co\,  o<:e<  I. 

n  +  l 

We  may  now  decompose  /[x)  into  two  parts 
Ax)=Mx)-\-f,{x),' 
where 

.^J  Uy  —  X  Uy  —  X        ,^mJ  ay  —  X 

1  «  +  I 

The  rational  function  y^(^)  has  poles  of  mod.  >  R  and  may 
therefore  be  developed  in  a  series  going  according  to  ascending 
powers  of  x^  convergent  for  |^  I  =  R'  >  R.     As  to  f^x^  we  ha\e 

(3)  /,(;c)  =  B„  +  B.a--h...  +  B„;r«+  .... 

where 


that  is 


a,"  L tf,         a„  .^       \  ay/         J 
but  we  have  by  hypothesis 


«  +  i 


and  therefore    ^^"(—j 
than  ^1  ^ol ,  so  that  we  have 


n  +  i 


<    I. 


will   by  condition  be  made  less 


d 


>B„a,-<  (d+  I) 
We  may  now  put  the  general  term  of  /Jix)  into  the  form 
B-a.'i^  ): 


i6 

which  shows  that  the  series  (3)  is  divergent  if  \x\  ^  a^y^K. 
The  two  series  /[{x)  and  /^(x)  have  therefore  the  circle  of  con- 
vergence C,  —  q.  e.  d.  Suppose  now  that  the  points  a^  are  con- 
densed along  certain  lines  or  arcs  of  curves.  We  have  then  a 
line  of  essential  singularities.  Suppose  also  that  our  curve  L 
admits  of  a  normal  being  drawn  to  it  at  a  point  a^ ;  by  the 
theorem  just  proved  the  function  f{x)  is  uniform  and  analytic 
within  the  circle  C"  having  its  center  x  "  on  the  normal  drawn 

O  o 

to  the  point  a^^  (Fig-   i)?   and  containing  no  isolated  poles,  if 


Fig.  I. 

such  exist.  It  is  moreover  evident  that  any  circle  whatever, 
cutting  off  a  part  of  L  cannot  be  a  region  of  convergence  for 
f{x) ;  for,  if  so,  let  x^  be  the  center  of  such  a  circle  C.  Draw 
now  a  circle  C"  with  center  x^'  on  the  mormal  to  some  point  a 
on  1/  and  entirely  within  the  larger  circle.  Since  f{x')  is  an 
analytic  function  inside  of  C  by  supposition,  we  may  develop  it 
at  x^  in  a  powerseries  V{x  —  x^')  having  a  radius  of  convergence 
greaterthan  \x—x^'\  which  is  contrary  to  the  theorem  just 
proved. 

The  question  now  naturally  presents  itself,  what  happens  if 
the  point  a^  is  a  point-limit  not  belonging  to  the  ensemble  ? 
That  such  a  point  differs  somewhat  from  a  point  of  the  ensem- 
'ble  itself  we  have  already  had  occasion  to  observe  in  the  case  of 

functions  of  the  type  f{x)  =  '^^ discussed    in  the   first 

part  of  this  paper.     That  fix).,  as  far  as  the  analytical  character 


17 

of  the  function  is  concerned,  behaves  at  a^  exactly  as  if  this 
point  were  an  essentially  singular  point,  is  evident  from  the  fact 
that  at  such  a  point  the  function  is  not  expansible  in  a  conver- 
gent powerseries,  there  being  an  infinite  number  of  poles  in  the 
neighborhood  of  a^.  It  may  however  ver}'  well  happen,  and  we 
shall  show  this  later,  that  the  function,  together  with  all  its  de- 
rivatives, remain  finite  at  such  points.  As  to  the  corresponding 
powerseries  two  things  may  happen,  either  is  it  divergent,  in 
which  case  it  is  seen  that  the  point  does  not  differ  from  an  essen- 
tially singular  point,  or  it  is  convergent,  having  a  certain  finite  or 
infinite  radius  of  convergence.  In  this  case,  however,  the  power- 
series  no  longer  represents  the  function  f{x),  which  now  has 
ceased  to  be  monogenic  and  analytic,  although  it  is  still  finite  at 
the  point.  We  proved  the  existence  of  such  functions  in  the 
first  part  of  this  paper. 

After  these  preliminary^  remarks  we  shall  study  a  somewhat 
more  special  fonn  of  functions  of  the  type 


/'')=2'3F^/ 


which  we  considered  above  in  general.  We  shall  first  take  the 
case  in  which  the  ai's,  as  well  as  their  point-limits,  are  condensed 
on  lines  or  curves.  Such  lines  we  shall  call  essentially  singular 
lines  of  the  first  category.  If  point-limits  of  the  d's  are  con- 
densed on  a  line  in  such  a  way  that  no  point  of  the  ensemble  a^ 
is  situated  on  it,  we  shall  call  such  a  line  an  essentially  singular 
line  of  the  second  category.  Functions  having  such  singulari- 
ties present  somewhat  different  features,  as  has  already  been  in- 
dicated and  will  be  discussed  in  a  subsequent  part. 
Consider  the  ensemble 

ay  =  ^"■"'*,  V  =  1 ,  2,   ....  00  , 
where  s  is  an  incommensurable  fraction.     This  ensemble  is  con- 
densed on  the  unit  circle.'     Let  us  now  put 


/(x)=2' 


!  x  —  e" 


'  I  have  not  been  able  to  find  a  short  and   rigorous  proof  of  this.     The 
proof  I  have  found  is  too  lengthy  to  be  given  here. 


which  is  derived  from  (i)  by  putting  ^^  =  — -  and  s  =  — .    This 

function  is  uniform  and  analytic  within  the  unit  circle  and 
therefore  developable  in  a  powerseries  P(^),  but  does  not  admit 
of  being  continued  beyond  the  circumference  of  the  circle.  Now 
since  f{x)  is  also  finite  and  continuous  outside  of  the  unit  circle, 

we  may  develop  it  in  a  powerseries  P  f  —  \.     Further,  since  an 

analytic  function  is  defined  only  in  that  part  of  the  plane  which 
can  be  reached  by  analytic  continuation, /"(;r)  represents  differ- 
ent analytic  functions  within  and  outside  of  the  unit  circle,  or, 
as  we  say,  one  function  is  lacunary  for  the  part  of  the  plane  in 
which  the  other  exists. 

The  respective  developments  are 

n  =  o 

M  =  00 

n  =  o 

As  a  more  general  case  consider  the  function 

o 

admitting  the  two  concentric  circles  C^^  and  C,.^  of  radii  r  and 
r^  respectively  as  essentially  singular  lines.  V{x)  may  now  be 
represented  by  the  series 

00 

J  \\-^  I  y    J        \„n  +  l      "      yn-\-\) 

■*™^  1  1 

a 

00 

o 


\x\  <r„ 

1^1  >^. 

r.<|^|> 

t 

19 

The  first  and  second  developments  hold  in  the  region  inside 
C^,  and  outside  C^^  respectively,  while  the  third  holds  in  the 
circle-ring  formed  by  C^^^  and  C^^;  this  last  development  is 
nothing  but  Laurent's  series. 

A  still  more  general  function  of  this  kind  is 

|ll  =00         !>  ==-  00 


''(->=2  2^;^ 


V  =  I 


where  ^^  ^^^^f  ^^  ^"  absolute  convergent  series.    This  function 

has  an  infinite  number  of  concentric  circles  Cr^  with  radii  r ,  r^, 
. . . ,  r^  as  lines  of  singularities.  Consider  now  the  circle-ring 
fonned  by  two  concentric  circles  Crp  and  Crp  +  i-  We  may  rep- 
resent F(;i:)  in  this  ring  by  the  sum  of  two  powerseries  P(x)  and 

f( — J  in  the  same  way  as  before,  and  we  find 
where 

»  00  P  00 


P+  I  o  < 

Let  us  now  make  the  transformation 


^W  JA  +  I 


ir^e' 


-t(^+t)- 


which  transfonns  the  ;r-plane  into  the  ^-plane  in  such  a  way 
that  the  circles  having  their  origin  as  center  are  transformed 
into  confocal  ellipses  having  the  points  -|-  i  and  —  i  as  foci ; 
in  particular  the  unit-circle  is  transformed  into  the  line  -f  i  —  i. 
Since  to  a  given  value  of  2  there  corresponds  two  values  of  .r, 
we  must  have  two  sheets  in  the  ^-plane,  in  order  to  represent  all 
the  values  of  x  in  it.  Further,  the  space  outside  of  the  unit- 
circle  is  transfonned  into  the  upper  sheet,  while  the  space  inside 
is  transfonned  into  the  lower  sheet,  the  two  sheets  having  con- 
nection all  along  the  line  -|-  i  —  i. 


20 

Suppose  now  we  transform  the  function 


by  the  above  transformation,  we  get 


V^—  (z  +  i/^-— I) 

in  which  the  radical  has  the  sign  +  or  — .  Now  since  the 
unit-circle  is  an  essentially  singular  line  fory(;i:),  the  cut  +  1  — i, 
which  is  evidently  nothing  but  a  flattened  ellipse,  is  an  essen- 
tially singular  line  for  <^(-3'),  hence,  to  /{^)  defined  as  a  uniform 
analytic  function  within  the  unit-circle  we  have  a  corresponding 
function  (f)(x)  in  which  the  sign  of  the  radical  must  be  chosen  so 
as  to  make  1 2  +  iZ-s'^  —  i  I  <  i,  that  is,  all  the  values  of  z  in  the 
lower  plane  ;  and  to  /{:v)  defined  in  the  space  outside  of  the 
unit-circle  there  corresponds  a  function  ^^(2")  in  which  the  radical 
sign  must  be  chosen  so  as  to  make   \2  -\-  \/ z^ —  i  |  >  i,  that  is, 

^^{2)  exists  in  the  upper  sheet.     The  development  ^^h^nX^  which 

defines /"(jir)  within  the  unit-circle  is  now  transformed  into  a  series 

^  KJyZ  +  ]  >^  —  I )",  which  is  convergent  in  the  lower  sheet 

for  all  values  of  z  with  the  exception  of  the  values  of  z  along  the 
cut  I  —  I. 

In  the  same  way,  to  the  development  ^B„[  —  j   ,   which 

defines  y(;r)  outside  of  the  unit-circle  corresponds  a  development 

^  B„(-2'  +  \/ z"^  —  i)"~"?  which  is  a  convergent  in  the  upper  sheet 

outside  of  the  cut.     We  are  thus  able  to  develop  the  function  in 
the  neighborhood  of  a  rectilinear  essentially  singular  line. 
Consider  now  the  function 


21 

which  has  all  the  concentric  circles  with  radii   i,  2,  ...,  00, 

— ,  — ,  — ,  . .  • ,   — ,  as  essentially  singular  lines.     In  the  ^-plane 
234  00 

we  get  confocal  ellipses,  situated  in  such  a  way  that  an  ellipse 
in  one  sheet  coincides  with  an  ellipse  in  the  other.  Suppose 
now  we  develop  f{x)  in  the  ring  formed  by  two  circles,  say  the 
unit-circle  and  the  next  one  interior  to  it ;  we  find  for  the  cor- 
responding development  in  the  upper  sheet 

00  00 

I  I 

which  series  converges  for  all  points  included  by  the  cut  and  the 
nearest  ellipse,  provided  the  sign  of  the  radical  be  so  chosen  as 
to  make  \z  +  \  z'  —  1 1  >  i.  A  similar  development  holds  for 
the  lower  sheet  in  the  region  coincident  with  the  region  in  the 
upper  sheet.     Let  us  now  put^(j')  in  the  form 

ae 

I 

'^^  {Z  -f  y^^^^i)n'^^'' 
I 

where  C„  =  B^  -f-  A„.     But    >^ = — is  convergent  out- 

side  of  cut  and  the  part  ^A,,  [{z  +  y^z'  —  i)"  —{z—  y/z'—  i)"] 

has  no  cut  at  all,  since  it  converges  for  all  values  of  z  on  it  and 
vanishes  for  ^  =  ±  i.     We  may  therefore  put 

][{z)-<t>{z)=^,{z), 

where  V^,(^)  is  the  part  having  no  cut  and  where  (f>(z)  is  put  for 

the  series    > "  — f-  A  .    The  analytical  continuation 

.^(z  —  X/z'  —  i)"         •»  ^ 

of  ir^{z)  from  one  sheet  into  another  is  thus  made  possible ;  in 

fact  "^jiz)  is  convergent  in  the  part  of  the  ^-plane  included  by 


22 

the  two  coincident  ellipses  in  the  upper  and  lower  sheet. 
Operating  on  "^^{2)  exactly  as  we  did  with  <f)(z)  we  get 

where  '^Ji_2^)  has  no  cut  and  none  of  the  two  ellipses  nearest  to 
it  as  essentially  singular  lines.  Continuing  in  this  way  we 
finally  arrive  at  the  identity 

§{z)—<l>{2)  —<f>A^)  —^M «^-t(^)  =^k  +  i{z), 

where  ifk  +  i'^^  ^  function  which  is  holomorphic  in  the  whole  re- 
gion included  between  k  ^  \  ellipses  in  the  upper  sheet  and  the 
corresponding  ellipses  exactly  below  it.  The  form  of  the  func- 
tion y\ri^j^^{s)  is  easily  obtained.  We  only  need  to  develop  f{x) 
by  Laurent's  theorem  in  the  region  between  the  >Hh  and  >^  +  ist 
ellipse.     We  obtain 

00  00 

I  I 

the  first  series  on  the  right-hand  side  is  convergent  for  all  values 
of  X-,  satisfying  the  condition  \x\  <ik  -\-  i.  We  now  write  as 
before 

where  the  first  series  is  convergent,  if  >^  +  i  >  |ji:  |  >  7 ,  and 

the  second  series  is  convergent  outside  of  the  circle  C^,  Hence 
we  may  put 

fix)  -  <!>,  {X)  =^M)  =2  ^"'  (^"  ~  ^  )  • 

or,  transforming  into  the  ^--plane, 

^{2)—<f>{2)=il^(2)  =  ^  A^'CC^+l/y-'+i)"— (^— t/^"— i)«], 

where  "^/^^  + 1  is  a  holomorphic  function  in  the  region  between 
the  ellipses  specified  above  and  vanishes  at  the  two  branch- 
points. 


23 

We  have  thus  seen  that  the  function  f{x\  or  its  transfonn 
i>{z\  belongs  to  that  class  of  functions  which  can  be  put  in  the 
form  ^ 

where  ^x)  has  one  essentially  singular  line  C  and  "^{z)  a  uni- 

fonn  function  which  may  or  may  not  have  other  singular  lines, 

but  not  the  singular  line  C  or  any  part  of  it.     The  difference 

f{x)  —  <^(:r)  has  therefore  no  essentially  singular  line  C.    -^hat 

this  is  not  always  possible  is  easily  seen  from  an  example  given 

by  Borel, 

VKz)  =  ^{z)-\-^S'')  log^, 

where  ^{z)  has  the  negative  part  of  the  real  axis  as  essentially 
singular  line  and  ^j^z)  is  a  polynomial.  Although  Viz)  is  an 
analytic  and  one  valued  function  outside  the  cut,  it  is  not  possi- 
ble to  continue  it  across  this  line  by  subtracting  any  term  of 
the  right  side  from  the  left,  in  fact,  the  difference 

F(^)— <^(^)  =  <^,(2')Iog(z) 

is  no  longer  unifonn  ;  this  is  due  to  the  fact  that  F(-s')  is  made 
unifonn  in  an  artificial  way  by  presiding  ^{z)  with  a  singular 
line,  which  prevents  the  variable  ^^  aescrib«/a  complete  path 
around  the  origin. 

Another  interesting  question  here  suggests  itself :  Is  it  pos- 
sible to  pass  continuously  with  a  function  f{x)  across  any  of  the 
ellipses,  say  the  cut  i  —  i,  that  is,  from  one  sheet  to  another?  It 
is  evident  that  as  long  as  we  define  f{z)  as  a  uniform  analytic 
function  in  Weierstrass'  sense  no  such  passage  is  possible,  since 
the  path  of  the  variable  is  not  restricted  at  all,  provided  it  re- 
mains within  the  circle  of  convergence.  But  suppose  we  restrict 
the  variable  to  move  along  certain  paths ;  in  that  case  we  may 
formulate  the  question  thus:  Is  it  possible  to  pass  across  an  es- 
sentially singular  line  without  the  function  suffering  a  break  of 
continuity?  That  this  is  possible  has  been  proved  in  a  theorem 
by  Borel,  which  we  shall  state  here  without  proof. ' 

Consider  the  series 


'  See  Borel's  thesis,  "  Sur  Quelques  Points,  etc."     Paris,  1894. 


24 
'^  An 

in  which  A„  is  an  ensemble  of  points  that  may  be  condensed 
along  lines,  or  even  in  areas,  and  m„  has  an  upper  finite  limit  n. 
Suppose  now  that  a  convergent  series  of  positive  terms  ^u„  exists 
such  that  the  series 

^>^  |A„1 


U„n 


is  convergent.     In  order  to  effect  this  it  is  only  necessary  and 
sufficient  that  the  series  of  positive  terms 


2"" 


l/|A„| 


shall  be  convergent.  Having  now  given  two  points,  P  and  Q, 
which  do  not  coincide  with  a  point  A„,  nor  with  any  point-limit 
of  the  ensemble,  Borel  proves  that  zV  is  always  possible  to  join 
these  points  by  a  continuous  line  7,  such  that  the  series  considered 
shall  be  absolutely  and  uniformly  convergent^  as  well  as  all  its 
derivatives^  on  this  line. 

The  determination  of  the  line  7  must,  in  general,  be  effected 
for  each  individual  function  and  constitutes  the  most  difficult 
problem  in  the  application,  of  this  important  theorem. 

It  is,  however,  not  difficult  to  prove  that  in  the  case  of  the 
function  f{x\  that  we  have  been  dealing  with,  certain  lines 
through  the  origin  may  be  chosen  as  the  line  7,  and  that,  more- 
over, an  infinite  number  of  such  lines  exist.  In  fact,  the  possi- 
bility of  passing  through  the  unit  circle  without  break  of  con- 
tinuity will  only  depend  on  the  possibility  of  choosing  coefficients 
c^  in  such  a  way  that 


/<^"'=2'?r^ 


shall  be  absolutely  convergent  for  one  or  more  values  of  B  not 
coinciding  with  any  one  of  the  values  i,  2,  3,  .  .  .,  co  .     We  have 


25 

^  e^  —  e"      ^  cos6'— 


cos  V  -{-  i  (sin  6  —  sin  v) 


=  ^2 


Cy 


.   e-^v  .   d—v  ,  .  /      d-\-v  .    e—v' 

—  sin sin h  z  I  cos sin 


_  __  _i_'^ ^ 

~  2    .^  6 — V     •  +  ».-. 

Sin e~j~'  > 


2 

that  is,  we  must  have  the  series 

Sin 

2 

convergent  for  one  or  more  values  of  0.  In  the  first  place,  sup- 
pose ^  =  o  or  TT,  in  order  that  (i)  may  be  convergent  it  is  only 
necessary  to  put 

Cf,  =  Cy  sin  V 
where  c,'  is  a  series  of  quantities  satisfying  the  conditions 

lim.  -j^^—=i  I,  lim.  Cy  =  o. 

v^oo    ^y — I 


I'  =00 


These  conditions  being  satisfied,  the  series 

^f/sin  -^,    2'^"' cos  -^ 

are  absolutely  convergent  (See  Picard  Traits  d'An.,T.  I.,  p.  231) 
and  our  theorem  is  proved  for  two  values  of  0.  If  then  we  fol- 
low the  path  along  the  real  axis,  we  are  able  to  pass  continuously 
through  the  unit  circle  at  the  jx)ints  ±:  i.  In  general,  suppose 
we  choose  our  coefficients 


c,z=  Cy  sill  V  TT  a 


.     6  —  V       «r=-i,4,   i5,...,oo 
;„,  Sin  ^'  ^'  ^* 


2  w  =  I,  2,  3,  . . . ,  ;/  —  I 

9=J!!_2ir 
n 

where  ^c,'  is  an  absolutely  convergent  series.     The  product 

TTa«.«  sin 

6  = nr 


26 

will  be  absolutely  convergent  for  any  fixed  value  of  v  if  the 
series  ^  («,«,„  sin i)  enjoys  the  same  property.  Hav- 
ing chosen  the  coefficients  a,n,  „  so  as  to  satisfy  this  condition, 
we  form  the  function 

Cv  sin  V  I  I  a^n  sin 


^     n 


X  —  ^"  ' 

which  is  absolutely  convergent  at  the  points  ^  =  —  27r.  We  may 

n 

therefore  pass  continuously  with  f{pc)  through  the  circle  along 
lines  whose  directions  at  the  points  of  exit  pass  through  the 
origin  and  form  angles  with  the  real  axis  equal  to  any  commen- 
surable part  of  27r.  These  lines  will  in  the  ^--plane  be  repre- 
sented by  homofocal  hyperbolae  leading  from  one  sheet  into 
another. 

III. 

We  shall  now  consider  a  class  of  functions  having  essen- 
tially singular  lines  of  the  second  category.  We  defined  these 
as  lines  on  which  are  condensed  point-limits  of  the  ensemble  not 
belonging  to  the  ensemble  itself.  We  said  above  that  on  such 
lines  the  function  may  be  finite  and  continuous  as  well  as  all  its 
derivatives.  Prof.  Mittag-Leffler  in  a  note  in  Acta  Math.^  Vol. 
15,  called  the  attention  of  mathematicians  to  a  function  of  this 
nature  admitting  of  no  analytic  continuation  across  the  unit 
circle,  but  possessing  the  property  of  remaining  finite  and  con- 
tinuous on  this  line  as  well  as  all  its  derivatives.  Pringsheim 
in  a  paper  already  mentioned  discusses  a  class  of  functions 
which  seem  to  posses  a  similar  property.  He  starts  with  the 
expression 


/(^>=2 


ay 


where  the  ensemble  a^  has  its  point-limits  condensed  on  a  line  in 
such  a  way  that  all  the  poles  are  on  one  side  of  it ;  thus,  for  in- 


31 

Since  f{z)  is  fiolomorphic  within  the  region  l-s-K  i,  it  may 
be  dexeloped  in  a  convergent  powerseries 

Now  since  f{z)  is  to  be  identical  with  ^{z)  for  all  values  of  z^ 
such  that  \z\  <  R',  we  must  have 

(2)  :  : 

2  ^  ^  ^'" 


The  identity  J\z)  —  (f>{z)  within  the  region  \z\  <  R'  leads 
therefore  to  a  system  of  an  infinite  number  of  equations  with  an 
infinite  number  of  unknowns  which  must  be  satisfied  by  the  r's, 
and  these  are  thus  seen  to  be  functions  of  the  a's.  Conversely, 
if  solutions  of  (2)  exist,  it  will  be  possible  to  make  /{z)  equal  to 
any  given  holomorphic  function.  Now  we  do  not  know  in  gen- 
eral how  to  solve  such  a  system.     Poincar^'s  method  fails- of 

course,  since  here  lim. eqiials  some  finite  quantity,  while  it 

c  =  00      ^y 

ought  to  be  equal  to  00 .  That  in  certain  cases  solutions  do 
exist  seems  evident  when  we  consider   Phragin^n's   example, 

/(^)=^/»(^)>  where  /„{z)=^  u^^-z'  w^^^^^'  ^^^^^  ^^v^^" 
oped,  gives  rise  to  a  system  of  equations  similar  to  (2).  If  then 
the  coeflScients  are  not  solutions  of  (2),  we  may  be  sure  that  no 
ambiguity  can  arise,  so  that,  when  we  speak  of  developing  /{z) 
across  the  unit  circle,  we  mean  /{z)  and  no  function  that  can  be 
so  continued  and  which  is  identical  with  it  for  any  particular 
region.       The  possibility  that  /\z)    may  represent  any  given 


32 

function  in  one  part  of  the  plaiie  and  a  transcendental  function 
in  another  has  been  overlooked  by  Pringsheim.  There  can  be 
no  doubt,  however,  that  the  functions  which  he  discusses  do  not 
possess  this  property,  as  it  is  evident  that  a  series  of  quantities 
c^^  only  subject  to  the  condition  ^c^  =  a  convergent  series,  do  not 
in  general  satisfy  the  system  (2). 

Another  method  of  constructing  functions  possessing  the 
property  of  being  finite  and  continuous,  as  well  as  all  its  deriva- 
tives, on  the  unit  circle  has  been  given  by  Borel  who  starts 
with  the  function 

where  the  exponents  c^  fulfill  the  condition 

Cv 

N  being  a  fixed  quantity.  The  unit  circle  being  a  singular  line 
for  f{2)  (See  Hadamar's  paper,  Lionville,  Fourth  series,  T.  VII, 
p.  115),  we  may  always  choose  coefficients  b^  in  such  a  way  as  to 
render  <^{z)  and  all  its  derivatives  finite  and  continuous  on  the 
circle.  The  function  constructed  by  Fredholm  belongs  to  this 
type.  If  we  transform  such  a  function  by  means  of  the  well- 
known  transformation  z  =  e^^  and  take  the  real  part  of  it,  we 
obtain  a  new  function  f[6)  of  a  real  variable  to  which  Taylor's 
development  cannot  be  applied  in  spite  of  the  fact  that  fO)  and 
all  of  its  derivatives  are  finite  and  continuous  for  all  values  of 
the  variable.'  The  important  connection  that  thus  is  seen  to 
exist  between  certain  functions  of  an  imaginary  variable  and 
functions  of  a  real  variable  which  are  not  developable  at  any 
point  in  a  Taylor's  series  may  perhaps  justify  the  search  for 
other  transcendentals  possessing  the  same  property  as  the  type 
given  by  Pringsheim  and  Borel,  but  of  a  more  general  character, 
and  the  author  has,  he  believes,  succeeded  in  adding  a  new  and 
interesting  class  of  transcendentals  to  those  given  above. 

We  shall  call  a  point  an  essentially  singular  point  of  the 

'  For  proof  of  this  theorem  see  article  by  Pringsheim. 


33 

second  category^  if  it  is  a  point-limit  of  poles  of  a  function,  and 
we  shall  use  the  term  essentially  singular  point  of  the  first  category 
for  points  a  such  that  no  finite  power  (o  exists,  which  renders 
lim.  {z—d)^f{z)\\x\\ioxv[i\y  finite  whatever  be  the  path  alongwhich 

z  =  a 

z  tends  to  a.  If  points  of  the  second  category  only  are  condensed 
along  lines,  these  will  become  singular  lines  of  the  second 
category  by  our  former  definition  (p.  17)  and  if  points  of  the  first 
and  second  categories  are  condensed  on  lines,  these  will  become 
lines  of  the  first  category,  A  point  of  the  second  category  may 
be  a  point-limit  of  poles  as  well  as  of  essentially  singular  points. 
We  presuppose,  as  before,  that  no  point-limit  belongs  to  the  en- 
semble from  which  it  is  derived.  It  is  also  evident  that  whether 
the  singular  line  be  of  the  second  or  first  category,  it  will  affect 
the  function  in  the  same  way;  viz.^  prevent  any  analytic  con- 
tinuation beyond  it,  provided,  of  course,  the  line  be  closed. 
Consider  now  the  expression 

where  lim.  a^  =  o,  lim.  d^  =  o,  and  a  an  incommensurable  num- 

I*  =  00  K  =  00 

ber,  which  without  loss  of  generality  may  be  put  equal  to  — . 

The  quantities  a^  and  d^  being  supposed  positive,  the  ensembles 
(i  -f  a^Je"']  (i  -f  dy^"'  will  represent  points  outside  the  unit  circle 
approaching  it  asymptotically  as  v  increases  indefinitely.  The 
points  e"'  which  are  condensed  on  the  unit  circle  will  therefore 
be  point-limits  of  the  given  ensembles.  In  order  that  the 
product 

AA-'—  (I  -h  d,)e-'' 

shall  be  uniformly  convergent  it  is  necessary  and  sufficient  that 
the  series 

(3)  ^la^.-d"] 

be  absolutely  convergent.     The  ensembles  a,  and   d,  satisfying 


34 

this  condition,  the  product  (i)  will  represent  a  function  F(^) 
which  is  finite  and  continuous  throughout  the  whole  plane  with 
the  exception  of  the  points  ( i  +  b^e"'  and  the  points  e"^  on  the 
circle.  Within  the  unit  circle  V{z)  will  represent  a  uniform  and 
analytic  function  admitting  this  circle  as  a  singular  line.  This 
line  plays  a  double  role  with  reference  to  the  function  F(<2'),  the 
points  e""-  being  point-limits  of  poles  as  well  as  of  zeros ^  and  it  is 
evident  that,  if  we  draw  around  any  point  of  it  a  circle  however 
small,  an  infinite  number  of  poles  and  zeros  will  be  situated 
within  the  circle.  We  may  call  such  a  line  a  doubly  singular  line  of 
the  second  category.     As  a  particular  example,  suppose  we  put 

a^  =  4  ,  ^u=  — ^  which  make  ^\a,  —  b,\^^^^^  =  a.  con- 

vergent  series.     The  corresponding  function  will  be 

X    ^  —  (  I  H V  j^"' 

F(.)=n— ^ — ~ 

1   0—    I  +  — ,  W"* 

\  2V'/ 

and  will  have  the  properties  described  above.  An  infinite  num- 
ber of  ensembles  a^  and  b^  may  manifestly  be  formed  satisfying 
the  condition  (3).     Examples  of  such  ensembles  are 

av  ^=  — ,  bv  := ;  «v  =  —  ,   Oy  = — ,  etc. 

e"  e" — v'  2"  2"-^  I 

Remark.  —  It  should  be  noticed  that  the  form  of  the  ensem- 
bles (i  +  ^^),  (i  +  b^)  is  not  as  special  as  might  be  supposed ;  in 
fact,  the  ensemble  e"  is  for  our  purpose  just  as  appropriate,  since 
it    may  always   be    put  in  the  form   i  -f  <a;^,  a^  being  equal  to 

1 \- -f  •  •  •,  and  more  generally  a"'' , where  |«  |  >  i 

and  lim.  k^  =  o,  may  also  be  put  in  the  same  form. 

We  shall  now  take  up  the  study  of  functions  like  ^{2)  on 
the  singular  line  which  we  shall  suppose,  as  before,  to  be  the 
unit  circle  around  the  origin.     Let  then 

^     s        ''tt    2 —  (i  +  ay)e''^     ..  ,.        , 

F(^)  =    II  r — ; ^'  liui-  «v  =  o,    Inn.  b^  =  o, 


27 

stance,  we  may  put  a^=  (i  -\ j^',  the  poles  approaching  the 

unit  circle  in  a  narrowing  spiral.  Pringsheim  shows  that  it  is 
always  possible  to  choose  coefficients  c^  in  such  a  way  as  to  make 
/{z)  and  all  its  derivitives  finite  and  continuous  on  the  circle. 
In  fact,  we  have 

In  order  that  this  last  series  may  be  convergent  we  must  put 

(1)  kJ=  \cA{\aA  —  i), 

where  cj  are  the  terms  of  the  convergent  series  ^cj ^  and  in 
general,  since 

/'"''-'= '-"""-^  2  (z-l)-+- 

we  have 

that  is,  we  must  put 

(2)  \c,\=  k.'Ka,— 1)"+', 

where  ^Cy'  is  a  convergent  series.  This  condition  being  satis- 
fied the  condition  (i)  is  also  satisfied.  In  order  to  accomplish 
our  purpose  we  only  need  to  put  «  =  v  in  (2)  so  as  to  insure  the 

convergence  of  ^^  — _  "  ^^,  for  all  values  of  « ;  we  therefore 

put 

(3)  \c.\=  k/I(a.— i)H-t. 

so  that  the  function  /{z)  and  all  its  derivatives  up  to  any  order 
as  high  as  we  please  will  be  absolutely  convergent  on  the  unit 

circle,  if  the  r^'s  satisfy  the  condition   (3).     Thus,  suppose  we 

(I   \     .                                          I  /                   I  \  ""^ ' 
I  H j^' ;  we  also  make  \Cy\=  ^-  li  —  i J 

=  — i"  TX~i  and  we  have 


28 


i/<».(..»)i<«/2^  V='''2^^' 


a  convergent  series  for  all  values  of  n  as  large  as  we  please. 

An  objection  may  be  raised  against  some  of  Pringsheim's 
conclusions,  and  it  is  indeed  instructive  to  notice  how  very  care- 
ful we  must  be  in  drawing  conclusions  from  an  apparent  prop- 
erty of  a  function  represented  in  the  form  of  an  arithmetical  ex- 
pression.    Pringsheim  reasons  thus  :  a  function 


o 


/(-)=^. 


with  the  unit  circle  as  singular  line  admits  of  no  analytical  con- 
tinuation across  the  unit  circle,  since  in  the  neighborhood  of  any 


point  on  the  circle  there  is  an  infinite  number  of  poles  of  f{z). 
The  function  f{s)  is  therefore  two  distinct  functions  within  and 
outside  of  the  unit  circle.  But  this  is  not  necessarily  true.  In 
fact,  we  may  imagine  the  existence  of  a  function  of  the  same 
form  and  property  as  f{z)  for  which  an  analytical   continuation 


29 

is  possible.  That  such  a  function  can  be  formed  has  been  shown 
by  Prof.  Phragin^n,  who  proceeds  thus : ' 

Let  /{ti)  be  a  given  analytic  function,  holomorphic  within 
a  region  A.  Choose  within  A  an  infinite  number  of  circles  C„, 
such  that  C„  +  ,  lies  entirely  within  C„  and  let  C„  converge 
toward  a  given  circle  C,  which  may  be  represented  by  the  unit 
circle,  as  n  increases  indefinitely.  Choose  positive  quantities  e^, 
€^,  ...,€^   such    that    lim.  €^=:o.      Since /(«)  is  holomorphic 

I'  =  00 

within  A,  we  may  represent  this  function  in  the  region  witliin 
C,  by  Cauchy's  integral 

Jc. 
taken  along  C,.     We  may  therefore  choose  a  number  of  points 
w/  on  C,  and  corresponding  values  A/  ==/(«/ )(«\  +  ,  —  uj)  such 
that 

ever>'where  within  C^  differs  from  /{z)  by  less  than  e^,  that  is 

|/(^)— y,(^)|<£. 

In  the  same  way,  since  /{s)  —f^  {z)  may  likewise  be  represented 
by  a  Cauchy's  integral  taken  around  C^,  we  may  form  the  ex- 
pression 

A.(') 
z 


whose  poles  lie  on  C,  and  satisfy  the  inequality 

Continuing  in  this  way  we  may  form  new  expressions  f^z\  etc., 
and  we  have  finally 


(I)  /(^)=2''-^^ 


(^). 


'  Phragm^n's  method  of  fomiinj^  this  function  has  been  communicated  to 
me  in  a  letter  from  Prof.  Bjerkness  of  Kristiania  University. 


30 

an  expression  which  is  uniformly  convergent  within  C  and  rep- 
resents f{z)  in  that  region. 

It  would  be  easy  to  choose  our  expressions  such  that  it  con- 
verges even  outside  of  C,  leaving  out  the  poles  of  course.  If  cr^, 
cr^,  . .  . ,  (T^,  . .  . ,  be  a  series  of  positive  quantities  chosen  in  such 
a  way  that  2^^  converges,  we  may,  as  is  seen  by  comparing  with 
Cauchy's  integral,  choose /"^ (-2'), /"^(-s'),  . .  .,  in  such  a  way  that 

\fM  I  <  <^,  outside  of  C, 
\f{z)  I  <  o-.^  outside  of  C, 


1/^,  +  1(2')  I  <  o-^  outside  of  C„ 

and  from  these  inequalities  it  follows  that  the  expression  (i)  will 
be  uniformly  convergent  outside  of  C.  This  function  then  pos- 
sesses the  same  property  as  Pringsheim's,  but  we  know  that, 
since  f{z)  within  C  is  identical  with  the  given  holomorphic 
function,  it  can  be  continued  beyond  this  line.  This  seeming 
paradox  may  find  an  explanation  in  the  following  way.  Suppose 
we  have  a  function  holomorphic  in  a  given  region  which  we  may 
suppose  to  be  the  unit  circle  around  the  origin,  the  poles  all 
being  outside  of  it,  and  we  may  suppose  these  to  be  forming 
singular  lines  such  that  lim.  a^,^  i.     Let  this  function  berepre- 

sented  by  the  series 


/<^'=2.^ 


all  the  poles  being  of  multiplicity  i.  Suppose  further  that 
within  the  region  mentioned  the  following  equality  holds 

where  <^{2)  is  a  given  function  holomorphic  in  a  region  inclu- 
ding at  least  the  imit  circle.  This  function  may  be  put  in  the 
form 

<^{z)  =  A„  +  ^,z  +  A.y  +  . . .  +  A„2~  +  .  • . , 
convergent  for  \z\<  R',  R'>  R. 


35 

and  let  also  the  condition  (3)  be  fulfilled.     Putting  'F(z)  into  the 
form 

we  see  that  if  F(^)  shall  be  convergent  on  the  unit  circle,  that  is 
for  1^1  =  I,  we  must  have 

^  z—  {j-\'6^)g^' 
a  convergent  series  for  12-1  =  1;  but  we  have 

that  is,  we  must  have 

(4)  ^y    -^^^ —    =  an  abs.  conv.  series. 

If  then  this  condition  is  fulfilled,  F(2')  will  be  finite  and 
continuous  on  the  circle,  the  condition  (4)  including  the  condi- 
tion (3).  By  proper  choice  of  ensembles  we  may  satisfy  this 
condition ;  we  may  for  example  suppose  the  ensembles  a^  and  fiy 
written  in  the  form 

a,  —  —J,  by  =     ,        ,  ,,  lim.  a,' =  cc  , 
a„  civ  ±  Oy     V  ==  00 

in    which    case    our    condition    reduces    to   the   simpler   one 
^    -^  =:  a  convergent  series. 


av  =  — ,  Oy=: ;  Ak  =  — ,  Oy= ,  etc., 

e"  g'—i  2"  2"— I 


Examples  of  such  ensembles  are 

a„=  -— ,  by  = 
(r 

or,  choosing  the  form 

ay  ^=  aj  -}-  by,  by  :=  by, 

we  have  the  condition 

"ST*  o-y 
>^  —  =  a  convergent  series. 

We  may  thus,  for  instance,  put 


=  an  absolutely  convergetit  series^  shall  be  finite  and 


36 

1,1,  I  1,1,  I 

«.  =  —  +  -F ,   <>.  =  -o  ;  ay=  —  -\-—     ^-^  =  ^-  etc. 

e^  V  V  a  v'^  v^ 

\a  I  being  greater  than  unity  and  k  some  position  integer  >  i. 

We  have  thus  proved  the  theorem. 

//  is  always  possible  to  choose  ensembles  a^  and  b^,  such  that 
the  function 

where  a^,   and  b^   fulfil   the  condition  lim.  a^,  =  o^  lim.   b^=^o, 

>'  =  00,  1/  =  CO 

2j\  "bT^ 

continuous  within  as  well  as  on  the  unit  circle. 

A  remark  concerning  the  continuity  of  F(^)  on  the  unit 
circle  is  here  in  place.  It  is  evident  that  V{^s)  is  not  continuous 
for  points  on  the  unit  circle  in  the  ordinary  sense,  since  for  these 
points  no  coinplete  region  around  any  one  of  them  can  be  formed 
such  that  I  V{z  +  \z)  —  ^{z)  I  <  e.  This  inequality  can  only  be 
satisfied  for  a  region  around  the  point  lying  completely  within 
the  unit  circle  having  a  part  of  the  circle-line  as  boundary.  If 
the  variable  moves  within  this  region,  or  on  the  boundary  of  it, 
the  above  inequality  can  be  satisfied  by  taking  the  region  small 
enough.  We  may  therefore  say  that  Viz)  is  inwardly  continuous 
at  all  points  of  the  unit  circle. 

The  question  now  remains.  What  happens  to  the  derivatives 
of  V{z)  ?     Forming  the  first  derivatives  we  have 

FV)  =  F(.)  X  2  [,-(.  +  ,.),»  -.-(.1^,).^-]  = 

-f^.    .  y  ^ {a.  —  bv)e^^ 

^K^)  ^  2j  [^—  (i  H-a,)^-][^—  (I  -f-  b,)e^i-Y 

and,  since  F(^)  is  already  absolutely  convergent  for  |^|  =^  i,  in 
order  that  V{z)  shall  possess  the  same  property,  it  is  necessary 
and  sufficient  that 


2 


{ay  —  by^e" 


[2  — (i  -\^ay)e^^y^z—  (14-/^^)^"^ 


2 


< 

\av  —  bv\ 


[«.+  !  —  |^|J[^,+   I  —  Ul] 


37  ^ 

shall  be  absolutely  convergent  for  i^-l^i;    this  leads  to  the 
condition 


*5)  2A-7U 


an  abs.  con  v.  series, 


which  evidently  also  includes  the  condition  (4).  Let  us  now 
write  the  first  derivative  in  the  form 

we  find  for  the  second  derivative 

F"(^)  =  F'(2')<I»(^)  +  F(2r)<I.'(2'). 

The  condition  (5)  being  fulfilled,  F(2'),  ^{2)  and  V{2)  are  all 
finite  and  continuous  on  the  unit  circle,  and  it  is  therefore  only 
necessary  that  ^'{z)  be  absolutely  convergent  for  \2\^^i  in 
order  to  have  ^"{2)  enjoy  the  same  property.  This  leads  again 
to  the  following  condition 


<«>     2  [(.;)'- (7^)']=— 


conv.  series. 


If  this  condition  be  satisfied,  the  condition  (5)  and  (4)  will  mani- 
festly also  be  satisfied.  In  general,  forming  the  wth  derivative 
of  ^{z)  we  have 

F''(2')  =  F''-'(^)<l>(^)-|-(«— i)F''-»(2')*'(^)4-..-  +<!>''- •(^)F(2'). 
and  from  this  fonnula  we  see  that  if  the  derivatives  of  order 
lower  than  n  are  absolutely  convergent  for  |^|=i,  V{2)  will 
possess  the  same  property,  provided  also  <I>"~'(5')  be  absolutely 
convergent.     But  this  again  leads  to  the  condition 

If  this  condition  is  satisfied,  then  will  also  all  the  derivatives  of 
order  lower  than  n  be  finite  for  |2^|  =  i ;  in  fact,  if  condition  (7) 
is  satisfied,  then  will  also  the  series 


2[(z)-(xr]' 


where  r  is  any  positive  integer  less  than  ;;,  be  convergent ;  but 
this  is  nothing  but  the  condition  that  must  be  fulfilled  in  order 


38 

that  the  rth  derivative  of  F(^)  shall  be  absolutely  convergent  on 
the  unit  circle  q.  e.  d.     We  may  therefore  say, 

The  necessary  and  sufficient  condition  that  F{s)  together 
with  all  its  derivatives  up  to  the  «'*  order ^  n  being  a  positive  in- 
teger as  high  as  we  please^  shall  be  absolutely  convergent  on  the 
unit  circle  is 

y^^      { —  )  —  \~r)   J  ^^  an  abs.  conv.  series, 

a^  and  b^,  being  subject  to  the  conditions  lint,  a^  =  o,  lim.  b^  =  o. 

V  ^  00  y  =  00 

If  then  we  can  find  ensembles  a^  and  b^,  satisfying  the  above 
conditions  we  shall  have  solved  the  problem  of  constructing  a 
function,  admitting  the  unit  circle  as  doubly  singular  line  and 
such  that  the  function  itself  with  all  its  derivatives  up  to  any 
order  as  high  as  we  please  remain  absolutely  convergent  on  this 
line. 

Now  I  say  that 


2"     '  I 


are  such  ensembles.     In  fact,  we  have  lim.  a^  =  o,  lim.  b^  =  o, 

^.|"a^-»— /^,-""]  =  ^^["2""—  (2"—  ;^)""|    and    this    latter 

series  is  easily  seen  to  be  absolutely  convergent  for  values  of  « 
however  large.     We  have 

and  this  last  series  is  convergent  for  all  positive  integer  values 
of  n  however  large  ;  the  function 

F(^)=  n 


v=l 

Z  ■ 


(-^) 


39 

will  therefore  have  the  required  property.  The  condition  (7) 
reduces  to  a  simpler  form  if  we  write  the  ensembles  a^  and  b^  in 
the  form 


a,' '     "      a/  ±  V 
and  we  have 

This  last  series  must  therefore  be  convergent  for  values  of 
//  however  large,  and  this  may  obviously  be  effected  by  making 
a^'  and  d/  satisfy  the  condition 

(8)  ^a^'^  —  ^by  =  an  abs.  conv.  series. 
Examples  of  such  ensembles  are 

«/  =  y\  ^y=-Ty'  ^^—  2".  ^•''=-73;  «/  =  o.\  bj  =  ~, 

a  being  a  positive  quantity  >  i.     Again,   suppose  we  put  the 
ensembles  in  the  form 

we  have 

2r''"-ribi<2['"'"-''-'"(-x;)"]< 

which  last  series  must  be  absolutely  convergent;  that  is 

(9)  ^aK'^+'^i.'"' =1  an  abs.  conv.  series. 
Such  ensembles  are,  for  instance, 

a/  =  v',  */  =  v»''+4;  a/=2^  V=2•'»  +  *^  etc. 
We  have  thus  proved  the  following  theorem  : 

//  is  always  possible  to  choose  ensembles  a^  and  by  in  such  a 
way  that  the  /unction 

V(,\  —  TT  ^  —  ( '  +  a^)e^ 


40 

a7id  all  its  derivatives  up  to  atty  order  as  high  as  we  please  shall 
be  finite  and  continuous  within  and  on  the  unit  circle  around  the 
origin^  admitting  this  line  as  a  doubly  singular  line. 

The  function  ^{2)  being  analytic  within  the  unit  circle  may 
be  developed  in  the  neighborhood  of  an  ordinary  point  2'  lying 
in  this  region  in  a  powerseries  P(-3'  —  2').  If  2'  be  a  point  on  the 
unit  circle,  a  powerseries,  even  if  convergent,  will  manifestly 
cease  to  represent  the  function. 

The  special  property  of  V{2)  on  the  unit  circle  depends,  as 
we  have  seen,  on  the  choice  of  the  ensembles  a^  and  by.  The 
form  of  V{z)  is,  however,  rather  special,  the  points  of  discontinuity 
of  F(2')  being  simple  poles  and  point-limits  of  poles.  A  more 
general  function  admitting  the  unit  circle  as  a  doubly  singular 
line,  but  whose  points  of  discontinuity  are  essential  singularites 
and  point-limits  of  such,  may  be  constructed  as  follows  : 

Consider  the  expression 

z — (I  -h  av)i 
(I  -h  b,)e^ 
where 


here 

{a.  —  by^c'^  J   r     {a,—  bAe-^    "I  ' 

<^A2)- ^_^^  _^^^)^w-^2  L^_(i_^^^),,.d  -^ 


^^1  L^-  —  ( I  -|-  ^o^"'-! 


and  we  shall  suppose,  as  before,  lim.  a^  =  o,  lim.  b^,  =  o.  We 
shall  prove  that  ■>if{2)  is  a  holomorphic  function  everywhere  in 
the  plane  where  \2 — (i  +  b^e"^  |  >  |  «^  —  ^J  >  o,  and  in  particu- 
lar within  the  unit  circle.  In  establishing  this  proposition  we 
shall  adopt,  with  slight  modification,  a  method  of  proof  applied 
by  Picard  to  functions  of  a  similar  form.'  Denoting  the  general 
factor  of  -^X^)  t»y  u^  and  taking  the  logarithm  of  this  quantity, 
we  have 


'  See  Picard  :  Cours  d'Analyse,  T.  II.,  p.  145-149.  The  function  considered 
by  Picard  differs  from  the  above  in  having  the  essentially  singular  points  con- 
densed on  the  unit  circle  so  that  the  circle  becomes  an  essentially  singular  line 
of  the  first  category  and  not  of  the  second  category.  This  difference  is  all  im- 
portant, since  clearly  Picard's  function  can  present  no  such  property  as  the  one 
with  which  we  are  chiefly  concerned  in  this  paper. 


41 

1  —  1      g  —  ( I  +  a.)e''   ■       {a.  —  6.)e'"'      ,      r    (a.  —  d,)^'    "1  * 

'""^  "•""  ^°^^  — (I  +  ^J^- "^^— (I  +  *.)^' """^  L  — (I +*.)^'J 

,  _, I      r     {a,  —  b,)e-'     1"-' 

"•"  •"  "•"  V— I  Iz-ii-^  &,)€"' J        ' 

where  los: r — i — r\ — ;  is  a  uniform  function  within  the  unit 

circle.     Developing  we  find 

logu  -       ^  _  (  I  _j_  ^^)^..  1"  I  U  _  ( 1  ^4-  ^,)^w  J 

I      r    (a,  — ^.)g'"    1"-' I   r    {Uy—  dy)e'''    "I" 

V— I  L^  —  (i4-  du)e'"J  V   L^  —  ( I  -I-  ^,.)<"'' J 

"•"^  — (i  +  M^"'"^  •••• 
We  have,  therefore, 

log^(.)  =  -^-[^— -^^-^-^^J  + 

v4- 1  U  — (I  H-^^j^"'-!      "•"•••' 

from  which  we  derive 

i_  r     (g^  — ^.>)g'"    -I  "  + ' 

v+  I  u— (iH-*.')<?'"J        ~^  ' 

and  we  have  to  prove  now  that  the  series 

V  -I-  I  U  —  ( I  +  *^  W' J         "^  *  " 

is  finite  and  continuous  in  the  part  of  the  plane  where 
\2  —  (i  +  ^^)^'|>  \ay  —  by\  say  for  instance  the  unit  circle 
around  the  origin.     The  general  term  of  (ii)  is 

..  If     {a,  —  b,)e^'     Y 1      r     (a.—  b,)e^'    -]-  +  ' 


42 

Replacing  each  term  by  its  modulus  and  2  by  |  ^| ,  we  get 

(-)    :/'(I^1)I<v[t^1^,]  + 

[-      1^.— ^,1       y I 

^  Li  -\-dy—\2\  i       _      \a.  —  dy\       ' 

'  1+^.—    12-1 

But  the  series  whose  general  term  is  equal  to  the  right-hand 
side  of  (12)  is  absolutely  convergent ;  in  fact,  we  have 


lim     i  \     I  ^-^  ~  ^^  I      1 "_  Hm      l^"  —  ^-!     <-  I 

for  values  of  2  such  that  U  —  (i  +  d^)e'"'  \  <C\a^  —  dJ.  We  shall 
now  proceed  to  investigate  the  behavior  of  -^{s)  on  the  unit  cir- 
cle and  we  propose  to  find,  just  as  before,  the  necessary  and  suffi- 
cient condition  which  the  ensembles  a^  and  d^  must  fulfil  in 
order  that  '\/^(5')  and  all  its  derivatives  shall  be  finite  and  con- 
tinuous within  and  on  the  unit  circle. 
Putting  I  ^  I  =  I  in  (12)  we  have 

-a^  —  dy~\ " 
—  I 


(13)     i/.(^)i<-r-v^T    ;  , 

1^1  =  1      V         ]_  Oy  J  Uy    b^ 


I     


and  the  series  whose  general  term  is  equal  to  the  right-hand  side 
of  (13)  will  obviously  be  convergent  if 

which  therefore  is  the  condition  that  must  be  fulfilled  in  order 
that  ■>^{2)  shall  be  finite  for  1 5'  |  =  i.     Let  us  now  write 

where  4>{2)  is  identical  with  the  series  (11).     We  find  for  the 
successive  derivatives  of  "^{2) 


43 


4/(z)  =  ^(z)<l>'(z) 


ri2)  =  r-'{^)<t>'{z)+{n—i)r-H2)<t>"(2)-\-"-+^(2)<t>''{^)- 

These  expressions  show  that,  if  ylr{2)  be  finite  on  the  unit  circle, 
the  successive  derivatives  of  yfr{z)  will  have  the  same  property 
provided  also  we  can  make  the  successive  derivatives  of  (f>{z) 
finite  on  this  line.  Hence  to  find  the  condition  which  must  be 
satisfied  in  order  that  yfr{z)  as  well  as  all  its  derivatives  up  to  any 
order  as  high  as  we  please  shall  be  finite  and  continuous  inwardly 
on  the  unit  circle  is  manifestly  equivalent  to  finding  the  condi- 
tion that  <f>(z)  shall  possess  the  same  property. 
We  now  have 

^^    '      .^—{i-{-6y)e^'lz—{i-^6ne'"J 

which  series,  since  the  part  in  bracket  is  itself  a  convergent 
series,  evidently  can  be  put  into  the  form 

z—(i-\-dy)e^ 
In  the  same  way  we  find 


I 


and  in  general 


rv.v-\-  I  ...  v-\-n—  2 


44 


(n  —  i) 


V.  V  -\-  I   •  •  '  V  -{-  n  —  3 


n  —  II 


l2—{i-\-b,))e^y-^lz—{i-[-a,)e' 

Replacing  in  this  series  each  term  by  its  modulus  and  putting 
^^1^1  =  1  we  have 


i<^"(^)ii.i=i<2" 


I  a^ —  by]"      I     Vv.  V  -\-  I  .  .  '  V  -{-  n  —  2 


-r 


+  («-!) 


by"  Uy   L  by""- 

V.  V  -|-  I  ...  V  -|-  n —  3 
bj"  ^  ^ay 


+ 


«  —  1 1   1 


If  in  this  series  we  replace  each  numerical  coefficient  of  the 
terms  in  bracket  by  the  greatest  of  them,  viz.^  v.  v  +  i  ...  v  -t- 
n  —  2,  we  get 

Uy by    I         I 


+ ... 


<r 


^zil!2 


by^'-^Uy 

-\ (v.  v+i...    V  -\-  n  —  2) 

"'-"'''■[(f)-(^)"]J. 


since 


-\- 11  —  2I 


V —  I 


<C      V   I       M  ll 


In  order  that  this  last  series  shall  be  absolutely  convergent,  we 
must  have 


lim. 


V/(l:)-a)V/^l<- 


but  we  have 


and  therefore 


my-{~y\<{m' 


45 


litn. 


a,  —  by 


\/|\*,/       \aj 


<lim. 


we  have  also 


\/(z)" 


litn.  I  _^j  ==  lini.  ^  j  y/ 2ire 
and  we  find  the  condition 

lay  —  b, 


lim. 


Klii;)' 


V  I  ^lim. ! 


\l  \ay)    '  e 


<  I. 


which  will  be  satisfied  for  all  values  of  n  if  we  put  n  =  v.     We 
have  therefore  finally  the  condition 

dy  by 


(14) 


UyOy  V 


If  then  the  ensembles  a^  and  b^  be  chosen  in  such  a  way  as 
to  satisfy  the  condition  (14),  all  the  successive  derivatives  of 
<f>{s)  up  to  any  order  as  high  as  we  please  will  be  finite  on  the 
iniit  circle  and  the  relations  on  page  43  show  that  -^{2)  and  all 
its  derivatives  will  possess  the  same  property.  Examples  of  such 
ensembles  are 


(a)   ay  =:  — ,  Py  =^ ',  ay  =  — ,  Py  --= 

V  12"  .2 


,  etc. 


2"  + 


(b)    ay=     -,by= -^  .  ay  =  ~,by=~^^  +— ,  etc. 

If  we  write  the  ensembles  a^  and  b,  in  the  fonn 

<*"  fli>  ±  P|. 


the  condition  (14)  reduces  to 


bj<—. 


'  See  Serret :  Cours  de  Calcul  diff.  and  integ.,  T.  II,  p.  208. 


46 


which  is  seen  to  be  fulfilled  by  the  ensembles  (a)  given  above.  A 
second  form  of  ensembles  is 

—  —  A    —  -1  -h  -L 

Uy  Uv  Oy 


which  gives  us  the  condition 


I 


a^ 


=  ja/-a/(i=b^=F...  ) 


/2  "         e 


which  is  seen  to  be  satisfied  by  the  ensembles  {ti)  written  above. 
We  may  state  the  result  of  our  investigation  thus : 
//  is  always  possible  to  choose  ensembles  a^  and  b^  such  that 

the  function 


'AU)=n 


5'— (i  +  a,)^"'    ^y{z) 
e 


-(i-h  K)e^^ 

as  well  as  all  its  derivatives  up  to  any  order  as  high  as  we  please 
shall  be  finite  and  inwardly  continuous  within  and  on  the  unit 
circle  admitting  this  as  a  doubly  singular  line. 
We  found  above  the  following  inequality  : 


I^^C^)  I  <n  —  \\ 


Uv  —  bv 


1(i)-(t)j. 


\at,  —  bv 


and  we  notice  that  the  presence  of  the  factors  v  1  and 

is  due  entirely  to  the  fact  that  "^{s)  has  the  exponential  factor 
^^-'C^)  affixed.  If  this  factor  is  removed,  we  get  the  function  V{2) 
considered  above  ;  but  this  amounts  simply  to  putting  v  =  o  in 

these  factors,  since  o  I  =  i  and 


by 


=  I. 


Doing  this  we  get  the  inequality 

a  series  which  is  nothing  but  the  condition  obtained  on  p.  37 
for  F(^). 

If  we  study  the  ensembles  given  on  page  39,  we  notice  that 
the  poles  and  zeros  approach  each  other  more  rapidly  as  we  get 


47 

nearer  and  nearer  the  unit  circle.  On  the  other  hand,  the  series 
of  ensembles  on  page  45  ( (a)  and  (d) )  present  a  much  slower 
approach  of  zeros  and  essentially  singular  points.  It  is  not 
without  interest  to  notice  the  following  fact  : 

In  order  that  the  functions  F(^)  and  1/^(2')  and  all  their  de- 
rivatives shall  be  finite  and  continuous  on  the  imit  circle,  it  is 
necessary  and  sufficient  that  their  respective  logarithmic  deriva- 
tives shall  possess  the  same  property. 

Suppose  now  we  write  ^(2)  in  the  form 


Fu)=n(i-/^(^))-n(i '- 


5-—  (I  +  K)e'"' 


We  may  easily  prove  that  if  ^fviz)  and  all  its  derivatives  are 
finite  and  continuous  on  the  unit  circle,  the  same  will  hold  for 
F(5')  and  all  its  derivatives.  In  fact,  the  necessary  and  sufficient 
condition  for  this  is 


1—1^   '^  ^i^-(i -h *.)."" ('•+•  "^—'^ jui -(I -f^: 

<    «      >   - 


*►"  +  ' 


=  an  abs.  conv.  series. 


for  all  values  of  w  however  large.  But,  since  lim.  —j-  =  i,  the 
above  series  will  be  convergent  whenever  the  series 

is  convergent  and  conversely.  "Hence,  since  this  is  nothing  but 
the  condition  that  V{z)  shall  be  finite  on  the  imit  circle  as  well 
as  all  its  derivatives,  our  proposition  is  established. 

We  are,  by  means  of  this  proposition,  enabled  to  establish  a 
one-to-one  correspondence  between  the  types  of  functions  dis- 
cussed by  Pringsheim  and  functions  like  ^{2).  In  fact,  given  a 
function 


48 

where  ^^^  J^^  is  an  absolutely  convergent  series  for  all  values 
of  «,  if  we  write 

ay^=^bv-\-    Cy 


tlie  function 


>'w  =  n(.-7^?^f^.). 


which  has  the  same  poles  as  /{z)  and  whose  zeros  are  the  points 
of  the  ensemble  aj  =^  {i  -\r  b^  -\r  c^e"^  will,  by  the  above  propo- 
sition possess  the  same  property  as  f{z)  with  respect  to  the  unit 
circle.     Thus  suppose  we  start  with  the  function 


/(^)-^— 


^-(^+-7)^^'   ' 


we  put  <7^=  I  +  — H — -  making  the  series  ^  — ^^  absolutely 

convergent  for  all  values  of   n  and  the  corresponding  function 
F(^)  will  be 

F(-^)=  ri^-('+ v+^)^-- 


VITA. 

The  author,  John  Eiesland,  was  born  in  1867  near  Kris- 
tianssand,  Norway.  He  received  his  elementary  instruction  in 
the  public  school.  After  having  completed  in  1884  a  course  at 
a  normal  school,  he  attended  for  some  time  Kristianssand's  Latin 
and  Real  Skole.  He  afterwards  engaged  in  teaching  up  till  1888, 
when  he  came  to  the  United  States.  In  January,  1889,  he  en- 
tered the  State  University  of  South  Dakota,  from  which  insti- 
tution he  graduated  in  1891  with  the  degree  of  Ph.B.  From 
1 891-1892  he  taught  mathematics  and  sciences  in  a  private 
academy  in  Minnesota.  In  the  Fall  of  1892  he  entered  the 
Johns  Hopkins  University,  where  for  three  years  he  pursued  the 
study  of  mathematics  with  physics  and  astronomy  as  minor  sub- 
jects. 

In  1895  he  was  appointed  professor  of  mathematics  in 
Thiel  College,  Pa.,  which  position  he  still  holds,  having  a  leave 
of  absence  while  completing  his  course.  During  the  past  year 
he  has  held  a  university  scholarship  in  mathematics. 

In  conclusion  he  wishes  to  express  his  gratitude  to  Profes- 
sors Craig  and  Hulburt  for  the  interest  taken  in  his  work  while 
a  student  at  the  university. 


